JIN-YI CAI, Department of Computer Science, University of Buffalo,
Buffalo, New York 14260, USA

A new transference theorem in geometry of numbers with
applications to Ajtai's connection factor

Geometry of Numbers was christened by Minkowski as a synthesis of
geometric tools for number theoretic problems such as Diophantine
approximations and quadratic forms.

Not only are they fascinating, lattice problems may provide a new
class
of public-key cryptographic protocols usable in any communication
network, such as the internet and the World-Wide-Web. We will discuss
a recent breakthrough of Ajtai on a connection of average-case and the
worst-case complexity of lattice problems, and its implication for the
design of provably secure public-key cryptography.

We prove a new transference theorem in the geometry of numbers, giving
optimal bounds relating the successive minima of a lattice with the
minimal length of generating vectors of its dual, improving previously
the best transference theorem of this type due to Banaszczyk. We also
prove a stronger bound for the special class of lattices possessing
-unique shortest lattice vectors which play an important
role in the Ajtai-Dwork Crypto-system.

The theorems imply consequent improvement of the Ajtai connection
factors in the connection of average-case to worst-case complexity of
the shortest lattice vector problem.